Sharp Effective Finite-Field Nullstellensatz
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The (weak) Nullstellensatz over finite fields says that if P_1,…,P_m are n-variate degree-d polynomials with no common zero over a finite field 𝔽 then there are polynomials R_1,…,R_m such that R_1P_1+⋯+R_mP_m ≡ 1. Green and Tao [Contrib. Discrete Math. 2009, Proposition 9.1] used a regularity lemma to obtain an effective proof, showing that the degrees of the polynomials R_i can be bounded independently of n, though with an Ackermann-type dependence on the other parameters m, d, and |𝔽|. In this paper we use the polynomial method to give a proof with a degree bound of md(|𝔽|-1). We also show that the dependence on each of the parameters is the best possible up to an absolute constant. We further include a generalization, offered by Pete L. Clark, from finite fields to arbitrary subsets in arbitrary fields, provided the polynomials P_i take finitely many values on said subset.
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